Black hole size and phase space volumes

TL;DR

The paper argues that extremal black holes in string theory are best described as horizonless fuzzballs whose boundary areas scale with the enclosed microstate entropy, thereby linking geometric size to phase-space volume. By analyzing the NS1-P and its D1-D5 duals, Mathur shows that for the 2-charge system the boundary area satisfies a relation of the form $${A_E^{(10)}}/{G_N^{(10)}} \sim S_{micro}$, and extends the analysis to sub-ensembles where the entropy is reduced by a factor $M$ with a corresponding $1/M$ reduction in boundary area. It also applies the same reasoning to the rotating 2-charge black ring, obtaining $${A_E^{(10)}}/{G_N^{(10)}} \sim \sqrt{n'_1 n'_5 - J}$ for the full ensemble and $\sim (1/M)\sqrt{n'_1 n'_5 - J}$ for sub-ensembles. The work further analyzes quantum corrections, finding that string 1-loop effects ($R^4$ terms) remain small at the fuzzball boundary for a wide range of the sub-ensemble parameter $M$, supporting the robustness of the leading area-entropy relation. Altogether, the paper argues that phase-space considerations underpin the fuzzball geometry and suggests connections to holography and other entropy formalisms, potentially guiding understanding of black hole microstates in more general charge configurations.

Abstract

For extremal black holes the fuzzball conjecture says that the throat of the geometry ends in a quantum `fuzz', instead of being infinite in length with a horizon at the end. For the D1-D5 system we consider a family of sub-ensembles of states, and find that in each case the boundary area of the fuzzball satisfies a Bekenstein type relation with the entropy enclosed. We suggest a relation between the `capped throat' structure of microstate geometries and the fact that the extremal hole was found to have zero entropy in some gravity computations. We examine quantum corrections including string 1-loop effects and check that they do not affect our leading order computations.

Figures (6)

• Figure 1: (a) The 'naive geometry of the extremal hole (b) a special microstate geometry (c) A generic 'fuzzball' geometry.
• Figure 2: (a) The first drawing at the top depicts the NS1 string carrying a transverse vibration in a generic state; the string is opened up to its full length. Below that we draw the line traced out by this string in the noncompact directions; this curve lies in a ball shaped region given by the dotted line. Below that we sketch the geometry of the state: typical states of this type differ in the 'cap' region drawn by the dotted line, and we wish to find the area of the boundary indicated by the dotted circle. (b) The same as (a) but for a sub-ensemble of states which have shorter wavelength and smaller transverse displacements. The 'ball' is now smaller, and the throat deeper. We again wish to find the area of the boundary indicated by the dotted circle.
• Figure 3: (a) In the generic state with angular momentum $J$ the NS1 describes a path close to a circle; the tube shaped boundary encloses the region inside which the typical curve fluctuates (b) The paths for the sub-ensemble. The vibrations have smaller amplitude and higher wavenumber, and the tube is therefore 'thinner' with a smaller surface area.
• Figure 4: (a) The 2-charge geometry with the full ensemble of 'caps' (b) a sub-ensemble, which has deeper throats, and therefore a smaller boundary area for the fuzzball region (c) The naive geometry is taken to continue deeper than the deepest allowed cap. There are no states possible at the end of this throat.
• Figure 5: Length scales in the generic fuzzball. The curve has unit thickness, and the other length scales are larger by powers of $n$. In the generic state the location of the thick curve has quantum fluctuations that are large but these have been suppressed (by choosing high occupation numbers of modes) to enable an estimate of the string 1-loop effects.